Thomas Kerler – författare

d + 1-dimensional manifold, whose is a union of d-dimensional boundary disjoint v manifolds and d, a linear : -+ The manifold -Zod V(Md+l) V(Zod) V(Zld). ma- is with the orientation. The axiom in that z0g, Zod opposite gluing [Ati88] requires if we two such d + 1-manifolds a common d-subma- glue together along (closed) fold of in their the linear for the has to be the boundaries, composite compo- map tion of the linear of the individual d + 1-manifolds. maps the of and as in we can state categories functors, [Mac88], Using language axioms as follows: concisely Atiyah's very Definition 0.1.1 A in dimension d is a ([Ati88]). topological quantumfield theory between monoidal functor symmetric categories [Mac881 asfollows: V : --+ k-vect. Cobd+1 finite Here k-vect denotes the whose are dimensional v- category, objects for field tor over a field k, which we assume to be instance, a perfect, spaces The of of characteristic 0. set between two vector is morphisms, simply spaces the set of linear with the usual The has as composition. category Cobd+1 maps manifolds. such closed oriented d-dimensional A between two objects morphism.Zd d oriented d 1-- d-manifolds and is a + 1-cobordism, an + Zod meaning gMd+l = Zd is the d- mensional manifold, Md+l, whose Lj boundary _ZOd of the d-manifolds. consider union two we as joint (Strictly speaking morphisms cobordisms modulo relative Given another or homeomorphisms diffeomorphisms).

This book reviews recent results on low-dimensional quantumfield theories and their connection with quantum grouptheory and the theory of braided, balanced tensorcategories. It presents detailed, mathematically preciseintroductions to these subjects and then continues with newresults. Among the main results are a detailed analysis ofthe representation theory of U (sl ), for q a primitiveroot of unity, and a semi-simple quotient thereof, aclassfication of braided tensor categories generated by anobject of q-dimension less than two, and an application ofthese results to the theory of sectors in algebraic quantumfield theory. This clarifies the notion of "quantizedsymmetries" in quantum fieldtheory. The reader is expectedto be familiar with basic notions and resultsin algebra.The book is intended for research mathematicians,mathematical physicists and graduate students.